On the modelling and management of traffic
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 5, pp. 853-872.

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases are dealt with in detail through ad hoc numerical integrations. These are here obtained implementing the wave front tracking algorithm, which appears to be very precise in computing, for instance, the exit times.

DOI : https://doi.org/10.1051/m2an/2010105
Classification : 35L65,  90B20
Mots clés : optimal control of conservation laws, constrained hyperbolic pdes, traffic modelling
@article{M2AN_2011__45_5_853_0,
author = {Colombo, Rinaldo M. and Goatin, Paola and Rosini, Massimiliano D.},
title = {On the modelling and management of traffic},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {853--872},
publisher = {EDP-Sciences},
volume = {45},
number = {5},
year = {2011},
doi = {10.1051/m2an/2010105},
zbl = {1267.90032},
mrnumber = {2817547},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010105/}
}
Colombo, Rinaldo M.; Goatin, Paola; Rosini, Massimiliano D. On the modelling and management of traffic. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 5, pp. 853-872. doi : 10.1051/m2an/2010105. http://www.numdam.org/articles/10.1051/m2an/2010105/

[1] D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4 (1997) 1-42. | MR 1433310 | Zbl 0868.35069

[2] D. Amadori and R.M. Colombo, Continuous dependence for 2×2 conservation laws with boundary. J. Differ. Equ. 138 (1997) 229-266. | MR 1462268 | Zbl 0884.35091

[3] F. Ancona and A. Marson, Scalar non-linear conservation laws with integrable boundary data. Nonlinear Anal. 35 (1999) 687-710. | MR 1663615 | Zbl 0919.35081

[4] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609-645. | MR 2658157 | Zbl 1196.65151

[5] A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916-938. | MR 1750085 | Zbl 0957.35086

[6] C. Bardos, A.Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017-1034. | MR 542510 | Zbl 0418.35024

[7] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic. SIAM J. Appl. Math. (to appear). | MR 2765651 | Zbl 1217.35116

[8] A. Bressan, Hyperbolic systems of conservation laws - The one-dimensional Cauchy problem,Oxford Lecture Series in Mathematics and its Applications 20. Oxford University Press, Oxford (2000). | MR 1816648 | Zbl 0997.35002

[9] W. Chen, S.C. Wong, C.W. Shu and P. Zhang, Front tracking algorithm for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadratic, continuous, non-smooth and non-concave fundamental diagram. Int. J. Numer. Anal. Model. 6 (2009) 562-585. | MR 2574752

[10] R.M. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002) 708-721. | MR 1951956 | Zbl 1037.35043

[11] R.M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654-675. | MR 2300671 | Zbl 1116.35087

[12] R.M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow. Appl. Math. Lett. 17 (2004) 697-701. | MR 2064183 | Zbl 1058.49011

[13] R.M. Colombo, P. Goatin, G. Maternini and M.D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Transport Management and Land-Use Effects in Presence of Unusual Demand, L. Mussone and U. Crisalli Eds., Atti del convegno SIDT 2009 (2009). | Zbl 1211.35182

[14] R.M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions. J. Hyperbolic Differ. Equ. 7 (2010) 85-106. | MR 2646798 | Zbl 1189.35176

[15] R.M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. 70 (2010) 2652-2666. | MR 2678055 | Zbl 1211.35183

[16] C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33-41. | MR 303068 | Zbl 0233.35014

[17] C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28B (1994) 269-287.

[18] F. Dubois and P. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93-122. | MR 922200 | Zbl 0649.35057

[19] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44 (2006) 287-303. | MR 2239057 | Zbl 1134.35379

[20] J. Goodman, Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. Ph.D. thesis, California University (1982). | MR 2632689

[21] H. Greenberg, An analysis of traffic flow. Oper. Res. 7 (1959) 79-85. | MR 101166

[22] B. Greenshields, A study of traffic capacity. Proceedings of the Highway Research Board 14 (1935) 448-477.

[23] B. Haut, G. Bastin and Y. Chitour, A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions, in Proceedings 16th IFAC World Congress, Prague, Czech Republic, July (2005) Tu-M01-TP/3.

[24] D. Helbing, S. Lämmer and J.-P. Lebacque, Self-Organized Control of Irregular or Perturbed Network Traffic, in Optimal Control and Dynamic Games, Advances in Computational Management Science 7, Springer (2005) 239-274. | Zbl 1142.90350

[25] J.C. Herrera and A.M. Bayen, Incorporation of lagrangian measurements in freeway traffic state estimation. Transp. Res. Part B: Methodol. 44 (2010) 460-481.

[26] H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences 152. Springer-Verlag, New York (2002). | MR 1912206 | Zbl 1006.35002

[27] W.-L. Jin, Continuous kinematic wave models of merging traffic flow. Transp. Res. Part B: Methodol. 44 (2010) 1084-1103.

[28] W.L. Jin and H.M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model. Transp. Res. B 37 (2003) 207-223.

[29] W.L. Jin and H.M. Zhang, On the distribution schemes for determining flows through a merge. Transp. Res. Part B: Methodol. 37 (2003) 521-540.

[30] B.S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow. Phys. Rev. E 48 (1993) R2335-R2338.

[31] B.S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow. Phys. Rev. E 50 (1994) 54-83.

[32] B.S. Kerner and H. Rehborn, Experimental features and characteristics of traffic jams. Phys. Rev. E 53 (1996) R1297-R1300.

[33] A. Klar, Kinetic and Macroscopic Traffic Flow Models. School of Computational Mathematics: Computational aspects in kinetic models, XXth edition (2002).

[34] S.N. Kružhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | Zbl 0202.11203

[35] L. Leclercq, Bounded acceleration close to fixed and moving bottlenecks. Transp. Res. Part B: Methodol. 41 (2007) 309-319.

[36] L. Leclercq, Hybrid approaches to the solutions of the Lighthill-Whitham-Richards model. Transp. Res. Part B: Methodol. 41 (2007) 701-709.

[37] H. Lee, H.-W. Lee and D. Kim, Empirical phase diagram of traffic flow on highways with on-ramps, in Traffic and Granular Flow '99, M.S.D.W.D. Helbing and H.J. Herrmann Eds. (2000). | Zbl 0997.90523

[38] R.J. Leveque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002). | MR 1925043 | Zbl 1010.65040

[39] J. Li, Q. Chen, H. Wang and D. Ni, Analysis of LWR model with fundamental diagram subject to uncertainties, in TRB 88th Annual Meeting Compendium of Papers, number 09-1189 in TRB (2009) 14.

[40] M.J. Lighthill and G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955) 317-345. | MR 72606 | Zbl 0064.20906

[41] H.X. Liu, X. Wu, W. Ma and H. Hu, Real-time queue length estimation for congested signalized intersections. Transp. Res. Part C 17 (2009) 412-427.

[42] S. Mammar, J.-P. Lebacque and H.H. Salem, Riemann problem resolution and Godunov scheme for the Aw-Rascle-Zhang model. Transp. Sci. 43 (2009) 531-545.

[43] G. Newell, A simplified theory of kinematic waves in highway traffic, part II. Transp. Res. B 27 B (1993) 289-303.

[44] E.Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4 (2007) 729-770. | MR 2374223 | Zbl 1144.35037

[45] B. Piccoli and M. Garavello, Traffic flow on networks - Conservation laws models, AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). | MR 2328174 | Zbl 1136.90012

[46] P.I. Richards, Shock waves on the highway. Oper. Res. 4 (1956) 42-51. | MR 75522

[47] D. Serre, Systems of conservation laws 1 & 2. Cambridge University Press, Cambridge (1999). | MR 1707279 | Zbl 0930.35001

[48] C. Tampere, S. Hoogendoorn and B. Van Arem, A behavioural approach to instability, stop & go waves, wide jams and capacity drop, in Proceedings of 16th International Symposium on Transportation and Traffic Theory (ISTTT), Maryland (2005).

[49] B. Temple, Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3 (1982) 335-375. | MR 673246 | Zbl 0508.76107

[50] E. Tomer, L. Safonov, N. Madar and S. Havlin, Optimization of congested traffic by controlling stop-and-go waves. Phys. Rev. E 65 (2002) 4. | MR 1920601

[51] M. Treiber, A. Hennecke and D. Helbing, Congested traffic states in empirical observations and microscopic simulation. Phys. Rev. E 62 (2000) 1805-1824.